Finite Group Databasebeta

← #222231

Subgroups — #222231

Order 512.

944 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3107C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic
3#22509C211yesmaximal, characteristic, complemented
4C2×#1593C211yesmaximal, characteristic, complemented
5C2×#1584C211yesmaximal, characteristic, complemented
6#9661C211yesmaximal, characteristic, complemented
7#9660C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic, complemented
9#8950C211yesmaximal, characteristic, complemented
10#8950C211yesmaximal, characteristic, complemented
11#8948C211yesmaximal, characteristic, complemented
12#8600C211yesmaximal, characteristic, complemented
13#8314C211yesmaximal, characteristic, complemented
14#8312C211yesmaximal, characteristic
15#8149C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18#3107—24no—
19#3107—24no—
20C2×#485C411yescharacteristic
21C2×#485C411yescharacteristic
22C2×#475C411yescharacteristic
23C2×#474C411yescharacteristic
24C2×#462C411yescharacteristic
25C2×#461C411yescharacteristic
26C2×#452C411yescharacteristic
27C2×#452C411yescharacteristic
28C2×#446C2211yescharacteristic
29C2×#446C2211yescharacteristic
30C2×#446C2211yescharacteristic
31C2×C8×Q8C2211yescharacteristic
32C2×C4×Q16C2211yescharacteristic
33C2×C4×Q16C2211yescharacteristic
34C2×C4×SD16C2211yescharacteristic
35C2×C4×SD16C2211yescharacteristic
36C2×#431C2211yescharacteristic
37#1985—21no—
38#1984—21no—
39#1984C2211yescharacteristic
40#1984C2211yescharacteristic
41#1955—21no—
42#1955—21no—
43#1954C2211yescharacteristic
44#1954C2211yescharacteristic
45#1798—21no—
46#1797—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1750—21no—
50#1749—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1593—24no—
54#1584—24no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×#41C2×C411yescharacteristic
72C22×Q16C2×C411yescharacteristic
73C22×Q16C2×C411yescharacteristic
74C22×SD16C2×C411yescharacteristic
75C22×SD16C2×C411yescharacteristic
76#533—22no—
77#533—22no—
78#533—41no—
79#533—41no—
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×#125—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×C22⋊Q8C2×C411yescharacteristic
86C4×#41—22no—
87C4×#41—22no—
88C4×#41—41no—
89C4×#41—41no—
90C2×C4×Q8—21no—
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2311yescharacteristic
93C2×C4×Q8C2311yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4C2×C411yescharacteristic
98C2×C4×D4C2311yescharacteristic
99C2×#117—21no—
100C2×#117—21no—
101C2×#117C2×C411yescharacteristic
102C22×C42—21no—
103C22×C42C2311yescharacteristic
104#485—24no—
105#485—24no—
106#475—24no—
107#474—24no—
108#462—24no—
109#461—24no—
110#452—24no—
111#452—24no—
112#446—22no—
113#446—22no—
114#446—22no—
115#446—41no—
116#446—41no—
117#446—41no—
118C8×Q8—22no—
119C8×Q8—41no—
120C4×Q16—22no—
121C4×Q16—22no—
122C4×Q16D414yes—
123C4×Q16D414yes—
124C4×SD16—22no—
125C4×SD16—22no—
126C4×SD16—22no—
127C4×SD16—22no—
128#431—22no—
129#431—22no—
130C2×#107C2×C411yescharacteristic
131C2×#107C2×C411yescharacteristic
132C2×#106C2×C411yescharacteristic
133C2×#106C2×C411yescharacteristic
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2×C411yescharacteristic
139C2×C4⋊C8C2×C411yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#103C2×C411yescharacteristic
147C2×#103C2×C411yescharacteristic
148C2×#103C2×C411yescharacteristic
149C2×#103C2×C411yescharacteristic
150C2×#102C2×C411yescharacteristic
151C2×#102C2×C411yescharacteristic
152C2×C22⋊C8—21no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—21no—
155C2×C22⋊C8—21no—
156C2×C22⋊C8—21no—
157C2×C22⋊C8—21no—
158C2×C22⋊C8—21no—
159C2×C22⋊C8C2×C411yescharacteristic
160C2×C22⋊C8C2×C411yescharacteristic
161C2×C22⋊C8D411yescharacteristic
162C2×C22⋊C8D411yescharacteristic
163C2×#97—21no—
164C2×#97—21no—
165C2×#97—21no—
166C2×C4×C8—21no—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336D411yescharacteristic
181#336D411yescharacteristic
182#335—22no—
183#334—22no—
184Q8⋊C8—22no—
185Q8⋊C8—22no—
186Q8⋊C8—22no—
187Q8⋊C8—22no—
188Q8⋊C8—41no—
189Q8⋊C8—41no—
190D4⋊C8—41no—
191D4⋊C8—41no—
192#322—22no—
193#322—22no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41—41no—
197C2×#41—41no—
198C2×#41—41no—
199C2×#41—41no—
200C22×Q8—21no—
201C22×Q8C22×C411yescharacteristic
202C22×Q8C22×C411yescharacteristic
203C22×Q8C22×C411yescharacteristic
204C22×D4—21no—
205C22×D4C22×C411yescharacteristic
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4C22×C411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4#4111yescharacteristic
212C2×Q16—22no—
213C2×Q16—41no—
214C2×Q16—41no—
215C2×Q16—41no—
216C2×Q16—41no—
217C2×Q16C22⋊C414yes—
218C2×SD16—22no—
219C2×SD16—22no—
220C2×SD16—22no—
221C2×SD16—22no—
222C2×SD16—41no—
223C2×SD16—41no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C411yescharacteristic
234C22×C8C2×D411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—41no—
241C4⋊Q8—41no—
242#125—42no—
243#125—42no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8—41no—
247C22⋊Q8—41no—
248C22⋊Q8—42no—
249C22⋊Q8—42no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—22no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—42no—
260C4×Q8—42no—
261C4×Q8—41no—
262C4×Q8C2×D412yes—
263C4×Q8C2×D412yes—
264C4×Q8C2×D412yes—
265C4×Q8C2×D412yes—
266C4×D4—22no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—41no—
272C4×D4—42no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276#117—22no—
277#117—42no—
278#117—42no—
279#117—42no—
280#117—42no—
281#117—41no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42C22×C411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42#4111yescharacteristic
314C2×C42C2411yescharacteristic
315#107—22no—
316#107C22⋊C414yes—
317#106—22no—
318#106—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331C4⋊C8—42no—
332C4⋊C8—42no—
333#103—22no—
334#103—22no—
335#103—22no—
336#103—22no—
337#103C22⋊C414yes—
338#103C22⋊C414yes—
339#102—22no—
340#102—22no—
341C22⋊C8—22no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344C22⋊C8—41no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8—81no—
350C22⋊C8—81no—
351C22⋊C8—81no—
352#97—42no—
353#97—42no—
354#97—42no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—42no—
366C4×C8C2×D412yes—
367C4×C8C2×D412yes—
368C24—21no—
369C24C4×D411yescharacteristic
370#41—44no—
371#41—44no—
372#41—44no—
373#41—44no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—81no—
393C2×Q8C2×C22⋊C412yes—
394C2×Q8C2×C22⋊C412yes—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4#13112yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4111yescharacteristic
445Q16—44no—
446Q16—44no—
447SD16—44no—
448SD16—44no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8C2×C22⋊C412yes—
489C2×C8C2×C22⋊C412yes—
490C2×C8#13712yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4C2×C22⋊C412yes—
510C4⋊C4C2×C22⋊C412yes—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C42—21no—
518C42—21no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—42no—
529C42—41no—
530C42—41no—
531C42C22×D411yescharacteristic
532C42C22×D411yescharacteristic
533C42H(5,𝔽2)11yescharacteristic
534C42H(5,𝔽2)11yescharacteristic
535C23—21no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×C4×D411yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8—22no—
554Q8—41no—
555Q8—41no—
556Q8—41no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—84no—
563D4—22no—
564D4—22no—
565D4—48no—
566D4—41no—
567D4—41no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—84no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4#44412yes—
616C2×C4C22×C22⋊C411yescharacteristic
617C2×C4#51811yescharacteristic
618C2×C4D4211yescharacteristic
619C2×C4D4211yescharacteristic
620C2×C4#54611yescharacteristic
621C2×C4#54611yescharacteristic
622C2×C4C2×#13111yescharacteristic
623C2×C4#56811yescharacteristic
624C2×C4#56811yescharacteristic
625C2×C4#56811yescharacteristic
626C2×C4C2×#13711yescharacteristic
627C2×C4#57511yescharacteristic
628C2×C4#57711yescharacteristic
629C2×C4#57811yescharacteristic
630C8—22no—
631C8—22no—
632C8—22no—
633C8—41no—
634C8—41no—
635C8—41no—
636C8—41no—
637C8—41no—
638C8—81no—
639C8—81no—
640C8—81no—
641C8—81no—
642C8—81no—
643C8—81no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—42no—
648C22—41no—
649C22—42no—
650C22—42no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22D4×C22⋊C411yescharacteristic
659C22#293011yescharacteristic
660C22#293311yescharacteristic
661C22C4×#13711yescharacteristic
662C22#294711yescharacteristic
663C22C2×#44411yescharacteristic
664C22#295711yescharacteristic
665C4—21no—
666C4—21no—
667C4—22no—
668C4—41no—
669C4—42no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—42no—
674C4—81no—
675C4—81no—
676C4—81no—
677C4#287911yescharacteristic
678C4#288011yescharacteristic
679C4#297611yescharacteristic
680C4#297811yescharacteristic
681C4#327011yescharacteristic
682C4#327111yescharacteristic
683C4#327711yescharacteristic
684C4#327811yescharacteristic
685C2—81no—
686C2—82no—
687C2#1794111yescharacteristic
688C2#1794911yescharacteristic
689C2#1800911yescharacteristic
690C2#1802111yescharacteristic
691C2#2240011yescharacteristic
692C2#2244511yescharacteristic
693C2#2245211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22509C2semidirect
4. C2×#1593C2semidirect
5. C2×#1584C2semidirect
6. #9661C2semidirect
7. #9660C2semidirect
8. #8952C2semidirect
9. #8950C2semidirect
10. #8950C2semidirect
11. #8948C2semidirect
12. #8600C2semidirect
13. #8314C2semidirect
15. #8149C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#31072
#225322
#225092
C2×#15932
C2×#15842
#96612
#96602
#89522
#89502
#89482
#86002
#83142
#83122
#81492